<p>Why is the physical world described so precisely by mathematics? Every time I think about this dilemma Im stymied. At the most basic level the connection is tenuous, math deals with symbolic logic: assume certain statements, show that other statements follow, while science deals with observable phenomena: hypothesize, test, refine. Granted, the roots of math are grounded in describing the physical world, but even the most abstract mathematical ideas seem usable by science(example: group theory and quantum mechanics). Anyways, I thought this would be interesting to discuss, Ill stop now and let wiser minds continue.</p>
<p>A lot of abstract math is created because new science requires it. So to say how much of a coincidence it is that abstract math describes science is like saying how much of a coincidence it is that earth is so well suited for us humans.</p>
<p>Wow, zeta. I’ve thought about this question a lot, in precisely those words. It is a big mystery to me also – especially since a lot of math ends up precisely describing some physical phenomena AFTER it is invented abstractly and uselessly. I’ll say more later, but I just wanted to quickly record my amazement that someone has thought about this in such a similar way.</p>
<p>mathwiz, I define abstract/pure math to be mathematics which is NOT created to describe the physical world. Thus to say “A lot of abstract math is created because new science requires it” is a silly contradiction of terms.</p>
<p>It’s a silly contradiction in terms only under your definition of abstract (not necessarily mathwiz’ exact meaning) but you did not explicitly state your definition in your original post.</p>
<p>Perplex has a point. I would say calculus is certainly abstract/pure math, but at least one of its inventors created it for practical reasons.</p>
<p>Sorry, point conceded. I just thought that my original spiel made it clear what kind of math I was talking about.</p>
<p>No problem - we’re all on the same page now.</p>
<p>I found a fairly applicable essay by R. W. Hamming: <a href=“http://www.lecb.ncifcrf.gov/~toms/Hamming.unreasonable.html[/url]”>http://www.lecb.ncifcrf.gov/~toms/Hamming.unreasonable.html</a>. I’m not sure I agree with this guy, but the essay is certainly interesting.</p>
<p>This is one of many reasons why I am a theist.</p>
<p>Could you explain what you mean by “the roots of math are grounded in describing the physical world” ?</p>
<p>I feel like any abstract mathematical topic could be applied in some type of physical setting. The whole concept of “abstraction” applies to taking some sort of event or idea and simplifying it so that you can work with it in the mathematical setting. </p>
<p>It doesn’t seem so amazing to me. You can probably ‘force’ any mathematical idea upon a physical setting and get results that follow suit. </p>
<p>Perhaps I don’t know enough.</p>
<p>Hm! Interesting perspective. I’ve also never been really amazed by math’s relationship to science, because I’ve always believed that mathematics is a science. I suppose this takes some measure of faith, but I believe that mathematics isn’t so much synthesis as it is discovery – relationships already exist, and mathematicians find them, hypothesizing, testing, and refining just as scientists do. (What happens if the theorem is found not to work for some member of the universe? It’s thrown out or modified until there is a better solution.)</p>
<p>So as for the relationship between math and science, do physical phenomena cause math? No – that’s absurd. Do preexisting maths cause physical phenomena? I don’t know. The fact that we don’t know the mechanism by which something happens does not imply that the something can’t happen. Or do(es) some outside factor(s) cause both math and science? Certainly they don’t simply coincide?</p>
<p>I think the majority of mathematicians are Platonists (correct me if I’m wrong, Ben. 
) So, more to ponder – do theorems exist that are provable, but not provable by humans? What does that imply for science?</p>
<p>Teeeehee. I see that Ben is a Theorist, since he described mathematics as having been invented and not discovered. This may prove to be interesting.</p>
<p>Define math and science… and then we can start talking. =)</p>
<p>fool, the point is that abstract mathematics, invented by humans, has been found to describe the physical world with little “forcing,” like a puzzle which just fits together (without the need for reshaping the basic pieces themselves) and results in a coherent picture.</p>
<p>frozen-tears’ first post in this thread makes it too easy :(</p>
<p>No, not too easy – it should be the opposite. It’s too simple – a subtle difference. :)</p>
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<p>Although I think frozen-tears view is correct for certain aspects or time periods in mathematics, I don’t believe it is applicable to pure math today. For the last 200 years or so, mathematicians have been attempting to solidify the foundations of math(as a branch of logic), mainly by restricting the number of axioms(or starting assumptions). Sure, there was a bit of trial and error, a little testing of the waters to find the most simple axioms that keep the desired structure, but that is mostly done with- the axioms are set(ok so if you bring up the axiom of choice, we can talk). </p>
<p>At this point, we have a self-consistent logical structure, where theorems are not tweaked to fit reality, but are either true or false(or both-stupid Godel). This means if a mathematical theorem doesn’t jive with reality, we don’t just throw it out or modify it as frozen-tears suggested.</p>
<p>So the question is, restated, why does a logical system based on such a simple starting point have describe reality so well? Why do the theorems about groups apply to molecules? Why do physicists always find such convenient mathematical structures like Calub-Yau shapes to hang their theories on? (I’ve got the feeling most of the posters aren’t mathematicians, come on Ben where are you?)</p>
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<p>I asked this question of the most eminent group theorist now alive, Michael Aschbacher. His answer was remarkably simple and exactly on target. “Because things like molecules have automorphism/symmetry groups in quite natural way.”</p>
<p>Think about it a while. It’s remarkably natural.</p>